Abstract

We present a theoretical analysis of a new one-dimensional laser-cooling scheme that was recently demonstrated on a beam of metastable 4He atoms. Both internal and translational degrees of freedom are treated quantum mechanically. Unlike semiclassical approaches, such a treatment can be applied to situations in which the atomic coherence length is of the same order of or larger than the laser wavelength, which is the case for atoms cooled below the one-photon recoil energy. We introduce families of states that are closed with respect to absorption and stimulated emission, and we establish the generalized optical Bloch equations that are satisfied by the corresponding matrix elements. The existence of velocity-selective trapping states that are linear combinations of states with different internal and translational quantum numbers is demonstrated, and the mechanism of accumulation of atoms in these trapping states by fluorescence cycles is analyzed. From a numerical solution of the generalized optical Bloch equations, we study in detail how the final atomic-momentum distribution depends on the various physical parameters: interaction time, width of the initial distribution, laser detuning, laser power, and imbalance between the two counterpropagating waves. We show that the final temperature decreases when the interaction time increases, so that there is no fundamental limit to the lowest temperature that can be achieved by such a method. Finally, possible extensions of this method to two-dimensional cooling are presented.

© 1989 Optical Society of America

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  1. P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
    [CrossRef] [PubMed]
  2. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).
  3. Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);see also Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
    [CrossRef] [PubMed]
  4. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
    [CrossRef] [PubMed]
  5. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, (1989).
    [CrossRef]
  6. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
    [CrossRef]
  7. J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985), and references therein.
    [CrossRef]
  8. G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).
  9. E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976);H. R. Gray, R. W. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
    [CrossRef] [PubMed]
  10. P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
    [CrossRef]
  11. J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, in Interaction of Radiation with Matter, a volume in honour of Adriano Gozzini (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48.
  12. V. G. Minogin and Yu. V. Rozhdestvenskii, Zh. Eksp. Teor. Fiz. 88, 1950 (1985) [Sov. Phys. JETP 61, 1156 (1985)].The theoretical treatment of these authors is valid only for atomic momenta p larger than the photon momentum ℏk since their Fokker–Planck equation is based on an expansion in powers of ℏk/p.
  13. Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.
  14. Closed families exist only when the two counterpropagating waves have polarizations such that they cannot both excite the same atomic transition |g, m〉 ↔ |e, m′〉. This is always the case for a σ+−σ−configuration because of angular-momentum conservation. In the particular cases of Jg= 1 ↔ Je= 0 and Jg= 1 ↔ Je= 1 transitions, closed families also exist when the two counterpropagating waves have orthogonal linear polarizations. This is easily seen by use of new bases of sublevels for g and e, such as {|g, m= 0〉, [|g,m=−1〉±|g,m=1〉]/2}. Using these new bases, we find that the two waves cannot excite the same transition. This explains why cooling by velocity-selective coherent population trapping has been also observed on the 2 3S1−2 3P1transition of 4He with the orthogonal linear configuration.4
  15. Ch. J. Bordé, in Advances in Laser Spectroscopy, F. T. Arrechi, F. Strumia, and H. Walther, eds. (Plenum, New York, 1983);S. Stenholm, Appl. Phys. 16, 159 (1978).
    [CrossRef]
  16. R. J. Cook, Phys. Rev. A 22, 1078 (1980).
    [CrossRef]
  17. C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Amsterdam, 1977), p. 1.For an extension of these equations including translational quantum numbers, see S. Stenholm, Appl. Phys. 15, 287 (1978).
    [CrossRef]
  18. In fact, the exact shape of H(u) is not important, provided that it has the correct width 2ℏk and it is normalized. We have checked that a constant value over 2ℏk[H(u)=1/2ℏk for −ℏk⩽u⩽ℏk] yields almost identical results after only a few fluorescence cycles. We have thus taken the constant form for H(u), simpler for the calculations, for all the interaction times longer than 10Γ−1.
  19. In an experiment like ours,4the atoms are allowed to fly a long distance without any interaction until they are detected. Excited atoms will then decay to one of the ground states, and the recoil of the corresponding photon has to be taken into account. The last term of Eq. (5.4) must then be convoluted by the kernel H(u) introduced in Section 4. Note that this amounts to a convolution of σee(p) by a function with width 2ℏk. In the case of a high light intensity (for which our calculation is still valid), σee(p) assumes values comparable with those of σ++(p) or σ−−(p), and this convolution will produce a widening of the narrow structures of σee(p). In the case of a weak intensity, this correction is negligible.
  20. These results are readily obtained by following the method presented in C. Cohen-Tannoudji, Metrologia 13, 161 (1977).
    [CrossRef]

1989 (2)

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

1988 (2)

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

1985 (2)

V. G. Minogin and Yu. V. Rozhdestvenskii, Zh. Eksp. Teor. Fiz. 88, 1950 (1985) [Sov. Phys. JETP 61, 1156 (1985)].The theoretical treatment of these authors is valid only for atomic momenta p larger than the photon momentum ℏk since their Fokker–Planck equation is based on an expansion in powers of ℏk/p.

J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985), and references therein.
[CrossRef]

1982 (1)

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

1980 (1)

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

1977 (1)

These results are readily obtained by following the method presented in C. Cohen-Tannoudji, Metrologia 13, 161 (1977).
[CrossRef]

1976 (2)

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976);H. R. Gray, R. W. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

Alzetta, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976);H. R. Gray, R. W. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Bagnato, V. S.

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Bordé, Ch. J.

Ch. J. Bordé, in Advances in Laser Spectroscopy, F. T. Arrechi, F. Strumia, and H. Walther, eds. (Plenum, New York, 1983);S. Stenholm, Appl. Phys. 16, 159 (1978).
[CrossRef]

Castin, Y.

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

Chu, S.

Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);see also Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985), and references therein.
[CrossRef]

These results are readily obtained by following the method presented in C. Cohen-Tannoudji, Metrologia 13, 161 (1977).
[CrossRef]

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Amsterdam, 1977), p. 1.For an extension of these equations including translational quantum numbers, see S. Stenholm, Appl. Phys. 15, 287 (1978).
[CrossRef]

J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, in Interaction of Radiation with Matter, a volume in honour of Adriano Gozzini (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Cook, R. J.

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

Dalibard, J.

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985), and references therein.
[CrossRef]

J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, in Interaction of Radiation with Matter, a volume in honour of Adriano Gozzini (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Gould, P. L.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Gozzini, A.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

Helmerson, K.

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Knight, P. L.

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Lafyatis, G. P.

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Lett, P. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Martin, A. G.

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Metcalf, H. J.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Minogin, V. G.

V. G. Minogin and Yu. V. Rozhdestvenskii, Zh. Eksp. Teor. Fiz. 88, 1950 (1985) [Sov. Phys. JETP 61, 1156 (1985)].The theoretical treatment of these authors is valid only for atomic momenta p larger than the photon momentum ℏk since their Fokker–Planck equation is based on an expansion in powers of ℏk/p.

Moi, L.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

Orriols, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976);H. R. Gray, R. W. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

Phillips, W. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Pritchard, D. E.

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Radmore, P. M.

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Reynaud, S.

J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, in Interaction of Radiation with Matter, a volume in honour of Adriano Gozzini (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48.

Rozhdestvenskii, Yu. V.

V. G. Minogin and Yu. V. Rozhdestvenskii, Zh. Eksp. Teor. Fiz. 88, 1950 (1985) [Sov. Phys. JETP 61, 1156 (1985)].The theoretical treatment of these authors is valid only for atomic momenta p larger than the photon momentum ℏk since their Fokker–Planck equation is based on an expansion in powers of ℏk/p.

Salomon, C.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Shevy, Y.

Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);see also Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Wallis, H.

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

Watts, R. N.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Weiss, D. S.

Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);see also Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Westbrook, C. I.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (2)

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, (1989).
[CrossRef]

J. Phys. B (2)

J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985), and references therein.
[CrossRef]

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Lett. Nuovo Cimento (1)

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976);H. R. Gray, R. W. Whitley, and C. R. Stroud, Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

Metrologia (1)

These results are readily obtained by following the method presented in C. Cohen-Tannoudji, Metrologia 13, 161 (1977).
[CrossRef]

Nuovo Cimento (1)

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento 36B, 5 (1976).

Phys. Rev. A (1)

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

Phys. Rev. Lett. (2)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. G. Minogin and Yu. V. Rozhdestvenskii, Zh. Eksp. Teor. Fiz. 88, 1950 (1985) [Sov. Phys. JETP 61, 1156 (1985)].The theoretical treatment of these authors is valid only for atomic momenta p larger than the photon momentum ℏk since their Fokker–Planck equation is based on an expansion in powers of ℏk/p.

Other (9)

Other proposals for getting temperatures below the recoil limit have been presented. It has been suggested that optical pumping in translation space might be used to cool the translational degrees of freedom by velocity-selective recycling in a trap.See D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIII, S. Svanberg and W. Persson, eds. (Springer-Verlag, Heidelberg, 1987), p. 68.

Closed families exist only when the two counterpropagating waves have polarizations such that they cannot both excite the same atomic transition |g, m〉 ↔ |e, m′〉. This is always the case for a σ+−σ−configuration because of angular-momentum conservation. In the particular cases of Jg= 1 ↔ Je= 0 and Jg= 1 ↔ Je= 1 transitions, closed families also exist when the two counterpropagating waves have orthogonal linear polarizations. This is easily seen by use of new bases of sublevels for g and e, such as {|g, m= 0〉, [|g,m=−1〉±|g,m=1〉]/2}. Using these new bases, we find that the two waves cannot excite the same transition. This explains why cooling by velocity-selective coherent population trapping has been also observed on the 2 3S1−2 3P1transition of 4He with the orthogonal linear configuration.4

Ch. J. Bordé, in Advances in Laser Spectroscopy, F. T. Arrechi, F. Strumia, and H. Walther, eds. (Plenum, New York, 1983);S. Stenholm, Appl. Phys. 16, 159 (1978).
[CrossRef]

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Amsterdam, 1977), p. 1.For an extension of these equations including translational quantum numbers, see S. Stenholm, Appl. Phys. 15, 287 (1978).
[CrossRef]

In fact, the exact shape of H(u) is not important, provided that it has the correct width 2ℏk and it is normalized. We have checked that a constant value over 2ℏk[H(u)=1/2ℏk for −ℏk⩽u⩽ℏk] yields almost identical results after only a few fluorescence cycles. We have thus taken the constant form for H(u), simpler for the calculations, for all the interaction times longer than 10Γ−1.

In an experiment like ours,4the atoms are allowed to fly a long distance without any interaction until they are detected. Excited atoms will then decay to one of the ground states, and the recoil of the corresponding photon has to be taken into account. The last term of Eq. (5.4) must then be convoluted by the kernel H(u) introduced in Section 4. Note that this amounts to a convolution of σee(p) by a function with width 2ℏk. In the case of a high light intensity (for which our calculation is still valid), σee(p) assumes values comparable with those of σ++(p) or σ−−(p), and this convolution will produce a widening of the narrow structures of σee(p). In the case of a weak intensity, this correction is negligible.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11, proceedings of the Eleventh International Conference on Atomic Physics, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989).

Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);see also Y. Shevy, D. S. Weiss, P. J. Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, in Interaction of Radiation with Matter, a volume in honour of Adriano Gozzini (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48.

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Figures (13)

Fig. 1
Fig. 1

Three-level Λ configuration. (a) Two degenerate ground sublevels g± are coupled to an excited level e0 by two counterpropagating σ+ and σ circularly polarized laser beams with the same frequency ωL the corresponding coupling matrix elements are K+/2 and K/2 in frequency units. (b) Closed family of states coupled by interaction with the two lasers. Each state is characterized by its internal quantum number and its linear momentum along Oz.

Fig. 2
Fig. 2

Couplings and level widths for the three states |e0, p〉, |ψC(p) 〉, and |ψNC(p) 〉 of the family (p). |ψC(p) 〉 is coupled to |e0, p〉by the laser (coupling matrix element K / 2). |ψNC(p) 〉 is coupled to |ψC(p) 〉 by the motion (coupling matrix element kp/M). As a result of these couplings, |ψC(p) 〉 and |ψNC(p) 〉 acquire finite widths Γ′ and Γ″, respectively (departure rates). Γ is the natural width of |e0, p〉.

Fig. 3
Fig. 3

Redistribution among families by spontaneous emission. Spontaneous emission of a photon with linear momentum u along Oz (wavy lines) can bring an atom from the family (p) (solid lines) to the family (puℏk) [dashed lines in (a)] or to the family (pu + ℏk) [dashed lines in (b)]. Each state is represented by a point with an abscissa equal to its atomic momentum along Oz and by its internal quantum number e0 (upper horizontal line) or g± (lower horizontal line). The label of a family is the atomic momentum of its excited state.

Fig. 4
Fig. 4

Expected shape (a) of the atomic-momentum distribution P ( p at z ) and (b) of the population in the noncoupled state |ψNC(p)|σ|ψNC(p)〉. The vertical solid lines indicate the positions of the Dirac functions representing the contribution of the atoms in |ψNC(p) 〉. The dashed vertical lines indicate the positions of the Dirac functions representing the contribution of atoms in |ψNC(p)〉. For atoms accumulated in noncoupled states |ψNC(p)〉 with p in a narrow range δp around p = 0 (b), the expected atomic-momentum distribution consists of twin peaks centered at ±ℏk, with the same shape and the same width δp (a).

Fig. 5
Fig. 5

Time evolution of the atomic-momentum distribution P ( p at z ). The dashed curves with half-width Δp0 = 3ℏk show the initial distribution. As the interaction time θ increases, the height of the double peak at ±ℏk (characterizing the new cooling process) increases, and its width decreases. Conditions for these figures: laser detuning δL = 0; Rabi frequencies of the atom laser coupling |K+| = |K| = 0.3Γ

Fig. 6
Fig. 6

Half of Fig. 5(d) with a different scale showing the diffusion of a fraction of the atoms toward large values of the momentum.

Fig. 7
Fig. 7

Atomic population in the noncoupled states |ψNC(p)〉 in the same situation as for Fig. 5(d). The peak height is twice as large, and the width is the same as in one peak of Fig. 5(d). At this scale, the population in |ψNC(p)〉 would not be visible.

Fig. 8
Fig. 8

Half-width of the peaks (initial half-width Δp0 = 3ℏk, laser detuning δL = 0): (a) Δp for various interaction times θ for a Rabi frequency K = 0.3Γ; (b) Δp as a function of the Rabi frequency K = K+ = K for an interaction time θ = 1000Γ−1. These results show that Δp in proportional to θ−1/2 and to K (dashed lines) and thus confirm relation (5.6) for θ large enough that the two peaks are well separated.

Fig. 9
Fig. 9

Atomic-momentum distribution for unbalanced laser beams. Same conditions as for Fig. 5(d) except for the Rabi frequencies: K+ = 0.3Γ; K = 0.2Γ.

Fig. 10
Fig. 10

Atomic-momentum distribution for various detunings. Same conditions as for Fig. 5(d) except for the detunings δL± Γ(a), corresponding to δL = 0, is the same as Fig. 5(d)]. Cooling is efficient for any sign of the detuning.

Fig. 11
Fig. 11

Accumulation of atoms in the peaks as a function of time. The height of the peak (a) is a measure of the maximum atomic density in the p space. (b) Shows the fraction of atoms in the peaks. Conditions are the same as for Fig. 5 except for the initial distribution (Δp0 = ℏk).

Fig. 12
Fig. 12

Configuration for two-dimensional velocity-selective coherent population trapping. (a) The J = 1 ↔ J = 1 atomic transition with the corresponding Clebsch–Gordan coefficients. (b) The three laser wave vectors and polarizations for which the state defined in Eq. (7.1) is trapping and velocity selective along Ox and Oz.

Fig. 13
Fig. 13

Closed family of states coupled by interaction with the lasers of Fig. 12(b). Each state is characterized by its internal and external quantum numbers.

Equations (51)

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1 2 [ exp ( ikz ) | g + exp ( ikz ) | g ]
( p ) = { | e 0 , p , | g , p k , | g + , p + k } .
σ e e ( p ) = e 0 , p | σ | e 0 , p ,
σ ± ± ( p ) = g ± , p ± k | σ | g ± , p ± k ,
σ e ± ( p ) = e 0 , p | σ | g ± , p ± k ,
σ ± e ( p ) = [ σ ± e ( p ) ] * .
H = H A + V ,
H A = P 2 2 M + ω 0 | e 0 e 0 | .
V = D E ( z , t ) ,
E ( z , t ) = ½ { + E + exp [ i ( k z ω L t ) + c . c . ] } + ½ { E exp [ i ( k z ω L t ) + c . c . ] } ,
K ± = d ± E ± , d ± = e 0 | ± D | g .
e 0 | + D | g + = e 0 | D | g = 0 ,
V = [ K + 2 | e 0 g | exp ( i k z ) + K 2 | e 0 g + | × exp ( ikz ) ] exp ( i ω L t ) + H . c .
exp ( ± i k z ) = p | p p k | ,
V = p [ K + 2 | e 0 , p g , p k | + K 2 | e 0 , p g + , p + k | ] exp ( i ω L t ) + H . c .
σ e ± ( p ) = σ e ± exp ( i ω L t ) , σ + ( p ) = σ + ( p ) , σ i i ( p ) = σ i i ( p ) ( i = + , , e ) .
[ d d t σ ( p ) ] Ham = i K + * 2 σ e ( p ) + c . c . , [ d d t σ + + ( p ) ] Ham = i K * 2 σ e + ( p ) + c . c , [ d d t σ e e ( p ) ] Ham = i K + * 2 σ e ( p ) + i K * 2 σ e + ( p ) + c . c , [ d d t σ e + ( p ) ] Ham = i ( δ L + k p M + ω R ) σ e + ( p ) i K 2 [ σ + + ( p ) σ e e ( p ) ] i K + 2 σ e + ( p ) , [ d d t σ e ( p ) ] Ham = i ( δ L k p M + ω R ) σ e ( p ) i K + 2 [ σ ( p ) σ e e ( p ) ] i K 2 σ + * ( p ) , [ d d t σ + ( p ) ] Ham = i K + * 2 σ e + ( p ) + i K 2 σ e * ( p ) + 2 i k p M σ e + ( p ) ,
| ψ NC ( p ) = K ( | K + | 2 + | K | 2 ) 1 / 2 | g , p k K + ( | K + | 2 + | K | 2 ) 1 / 2 | g + , p + k ,
| ψ C ( p ) = K + * ( | K + | 2 + | K | 2 ) 1 / 2 | g , p k + K * ( | K + | 2 + | K | 2 ) 1 / 2 | g + , p + k .
e , p | V | ψ NC ( p ) = 0 .
e 0 , p | V | ψ c ( p ) = 2 ( | K + | 2 + | K | 2 ) 1 / 2 exp ( i ω L t )
d d t ψ ( p ) NC | σ | ψ ( p ) NC = i k p M 2 K + K | K + | 2 + | K | 2 × ψ ( p ) NC | σ | ψ ( p ) C + c . c .
Γ = 2 K 2 / Γ .
Γ = 2 ( k p / M ) 2 Γ K 2 .
( k p M ) 2 < K 2 2 Θ Γ ,
p 2 k p p + 2 k .
[ d d t σ e e ( p ) ] sp = Γ σ e e ( p ) ,
[ d d t σ e + ( p ) ] sp = Γ 2 σ e + ( p ) ,
[ d d t σ e ( p ) ] sp = Γ 2 σ e ( p ) .
k + k d u H ( u ) = 1
Γ + = Γ / 2 .
[ d d t σ + + ( p ) ] sp = Γ 2 k + k d u H ( u ) σ e e ( p + k + u ) .
[ d d t σ ( p ) ] sp = Γ 2 k + k d u H ( u ) σ e e ( p + u k ) .
H ( U ) = 3 8 1 k ( 1 + u 2 2 k 2 ) .
g , p | σ | g , p .
e , p u | σ | e , p u .
| g , k = 1 2 [ | ψ NC ( 0 ) + | ψ C ( 0 ) ]
P + 0 ( p at z ) = P 0 ( p at z ) .
σ + + ( p ) = P + 0 ( p + k ) , σ ( p ) = P 0 ( p k ) , σ e e ( p ) = 0 , σ + ( p ) = σ e + ( p ) = σ e ( p ) = 0 .
d σ d t = ( d σ d t ) Ham + ( d σ d t ) sp ,
P ( p at z ) = σ + + ( p at z k ) + σ ( p at z k ) + σ e e ( p at z ) .
| ψ NC ( 0 ) = 1 2 [ | g , k | g + , + k ]
δ p M k Γ K θ .
Γ = ( K 2 / 2 ) Γ δ L 2 + Γ 2 4 .
δ = ( K 2 / 2 ) δ L δ L 2 + Γ 2 4 .
Γ = ( k p / M ) 2 Γ δ 2 + Γ 2 4 .
Γ = ( k p / M ) 2 Γ K 2 / 2 ,
| ψ NC ( P ) = 1 3 ( | g 1 , P k ê z + | g 0 , P + k ê x + | g + 1 , P + k ê ̂ z ) ,
( P k ê z ) 2 = ( P + k ê x ) 2 = ( P + k ê z ) 2 .
P ê z = P ê x = 0 .
{ p a t x = 0 p a t z = + k , { p a t x = 0 p a t z = k , { p a t x = + k p a t z = 0 .

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